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Let $\mathbb{F}_q$ be a finite field of characteristic not equal to $2$ or$3$. We compute the weight enumerators of some projective and affineReed-Muller codes of order $3$ over $\mathbb{F}_q$. These weight enumeratorsanswer enumerative questions about plane cubic curves. We apply the MacWilliamstheorem to give formulas for coefficients of the weight enumerator of the dualsof these codes. We see how traces of Hecke operators acting on spaces of cuspforms for $\operatorname{SL}_2(\mathbb{Z})$ play a role in these formulas.

Weight enumerators of Reed-Muller codes from cubic curves and their duals